03. Detailed Explanation of Structural Parameters: Discrete Blueprint of Spacetime

Introduction: First Step of Building Blocks—Blueprint

In Article 02, we established triple decomposition of parameter vector $\Theta =({\Theta }_{\text{str}},{\Theta }_{\text{dyn}},{\Theta }_{\text{ini}})$. Now we dive deep into first type of parameter: Structural parameter ${\Theta }_{\text{str}}$.

Popular Analogy: Architectural Blueprint Determines Skeleton of House

Imagine building a castle with LEGO blocks. Before starting, you need an architectural blueprint answering following questions:

Basic Questions:

1. How many blocks? (Number of lattice points $N_{\text{cell}}$)
2. What type is each block? (Cell Hilbert space ${\mathcal{H}}_{\text{cell}}$)
3. How are blocks connected? (Graph structure, neighbor relations)
4. Built on plane or circular base? (Topology and boundary conditions)

Advanced Questions: 5. Do blocks have special symmetries? (e.g., mirror symmetry, rotation invariance) 6. Are some positions unable to place blocks? (Defects, non-uniform structure)

Universe’s situation completely analogous:

Structural parameter ${\Theta }_{\text{str}}$ is universe’s “LEGO blueprint”, answering:

• How many “spacetime lattice points”? → Lattice set $\Lambda$
• What “internal structure” does each lattice point have? → Cell Hilbert space ${\mathcal{H}}_x$
• How are lattice points “connected”? → Graph structure $(\Lambda ,E)$
• Is universe “open” or “closed”? → Boundary conditions
• What symmetries exist? → Symmetry group $G$

This article will explain these in detail.

Part I: Construction of Lattice Set $\Lambda$

Simplest Case: Regular Rectangular Lattice

Definition 3.1 ($d$-Dimensional Rectangular Lattice):

$$\Lambda =\prod _{i=1}^d\{0,1,\dots ,L_i-1\}={\mathbb{Z}}_{L_1}\times \cdots \times {\mathbb{Z}}_{L_d}$$

Parameters:

• Dimension $d\in \{1,2,3,4,\dots \}$
• Lattice lengths in each direction $L_1,\dots ,L_d\in \mathbb{N}$

Total Number of Lattice Points: $$N_{\text{cell}}=\prod _{i=1}^d{L}_i$$

Example 1 (One-Dimensional Chain): $$\Lambda =\{0,1,\dots ,99\}={\mathbb{Z}}_{100}$$

• $d=1$, $L_1=100$
• Total lattice points: $N_{\text{cell}}=100$

Example 2 (Two-Dimensional Square): $$\Lambda =\{0,\dots ,999\}\times \{0,\dots ,999\}$$

• $d=2$, $L_1=L_2=1000$
• Total lattice points: $N_{\text{cell}}=10^6$

Example 3 (Three-Dimensional Cube): $$\Lambda =\{0,\dots ,L-1{\}}^3$$

• $d=3$, $L_1=L_2=L_3=L$
• Total lattice points: $N_{\text{cell}}=L^3$

Cosmological Scale (in Planck length units ${\ell }_p\sim 10^{-35}\text{m}$):

• Observable universe radius: $R_{\text{uni}}\sim 10^{26}\text{m}=10^{61}{\ell }_p$
• If three-dimensional cubic lattice: $L\sim 10^{61}$
• Total lattice points: $N_{\text{cell}}\sim (10^{61}{)}^3=10^{183}$

(But this exceeds $I_{\max }\sim 10^{123}$! Will discuss how to reconcile later)

Encoding Lattice Set

Encoded Content (part of ${\Theta }_{\text{str}}$):

1. Dimension $d$:

   • Use $\lceil {\log }_2{d}_{\max }\rceil$ bits
   • If restrict $d\le 16$ (sufficient for physics), need 4 bits

2. Lattice Lengths in Each Direction $L_1,\dots ,L_d$:

   • If each $L_i\le 2^{64}$, each needs 64 bits
   • Total: $64d$ bits

Bit Count: $$I_{\Lambda }=4+64d$$

Example ($d=3$): $$I_{\Lambda }=4+64\times 3=196\text{ bits}$$

Graph Structure and Neighbor Relations

Lattice set $\Lambda$ itself is just set of points. To define “which lattice points are neighbors”, need graph structure.

Definition 3.2 (Lattice Graph):

$${\mathcal{G}}_{\Lambda }=(\Lambda ,E)$$

where $E\subset \Lambda \times \Lambda$ is edge set, $(x,y)\in E$ means $x,y$ are neighbors.

Standard Choice (Rectangular Lattice):

(1) Nearest-Neighbor Graph:

$$(x,y)\in E\iff \sum _{i=1}^d|x_i-y_i|=1$$

(Manhattan distance equals 1)

Example (Two-Dimensional Square):

• Nearest neighbors of point $(5,7)$: $(4,7),(6,7),(5,6),(5,8)$ (4 neighbors)

(2) Next-Nearest-Neighbor Graph:

$$(x,y)\in E\iff 1\le \sum _{i=1}^d|x_i-y_i|\le 2$$

Example (Two-Dimensional Square):

• Next-nearest neighbors of point $(5,7)$: Besides 4 nearest neighbors, also 4 diagonal directions (total 8)

(3) Chebyshev Graph:

$$(x,y)\in E\iff \underset{i=1}{\overset{d}{\max }}|x_i-y_i|=1$$

Degree:

Number of neighbors (degree $\mathrm{deg}(x)$) for each lattice point:

• One-dimensional nearest neighbor: $\mathrm{deg}=2$ (interior points), $\mathrm{deg}=1$ (boundary points)
• Two-dimensional square nearest neighbor: $\mathrm{deg}=4$ (interior), $\mathrm{deg}\in \{2,3\}$ (boundary)
• Three-dimensional cube nearest neighbor: $\mathrm{deg}=6$ (interior)

Encoding Graph Structure:

For standard regular lattices, graph structure uniquely determined by neighbor type:

• “Nearest neighbor” → 1 bit encoding option
• “Next-nearest neighbor” → Another 1 bit
• Total: 2-3 bits

For non-regular graphs, need to encode adjacency matrix (expensive, usually avoided).

Physical Meaning: Lattice Points = “Pixels” of Spacetime Events

Classical Continuous Spacetime:

• Events: $(t,x,y,z)\in {\mathbb{R}}^4$ (continuous)
• Uncountably infinite points

Discrete QCA Spacetime:

• Events: $x\in \Lambda \subset {\mathbb{Z}}^d$ (discrete)
• Finite number of lattice points $N_{\text{cell}}$

Lattice Spacing $a$:

$$a=\frac{L_{\text{physical}}}{L_{\text{lattice}}}$$

Example:

• Physical length: $L_{\text{physical}}=1\text{m}$
• Number of lattice points: $L_{\text{lattice}}=10^{35}$ (in Planck length units)
• Lattice spacing: $a=10^{-35}\text{m}={\ell }_p$

Popular Analogy:

• Continuous spacetime = Photo with infinite resolution
• QCA lattice = Pixels of digital photo
• Lattice spacing $a$ = Physical size represented by each pixel

Part II: Cell Hilbert Space ${\mathcal{H}}_{\text{cell}}$

Internal Degrees of Freedom of Single Cell

Each lattice point $x\in \Lambda$ carries a finite-dimensional Hilbert space ${\mathcal{H}}_x$.

Definition 3.3 (Cell Hilbert Space):

$${\mathcal{H}}_x\cong {\mathbb{C}}^{d_{\text{cell}}}$$

where $d_{\text{cell}}\in \mathbb{N}$ is cell dimension.

Physical Meaning:

• $d_{\text{cell}}$: How many “internal quantum states” each lattice point has
• Analogy: How many color channels each pixel has (RGB = 3 channels)

Physical Origin of Cell Dimension

In real physics, ${\mathcal{H}}_{\text{cell}}$ usually decomposes as tensor product of multiple subsystems:

$$\boxed{{\mathcal{H}}_{\text{cell}}={\mathcal{H}}_{\text{fermion}}\otimes {\mathcal{H}}_{\text{gauge}}\otimes {\mathcal{H}}_{\text{aux}}}$$

(1) Fermion Degrees of Freedom ${\mathcal{H}}_{\text{fermion}}$

Simplest (Dirac-QCA): $${\mathcal{H}}_{\text{fermion}}={\mathbb{C}}^2$$

• Basis: $\{|\uparrow ⟩,|\downarrow ⟩\}$ (spin up/down)
• Dimension: $\dim {\mathcal{H}}_{\text{fermion}}=2$

Standard Model (3 generations leptons+quarks):

• Leptons: Electron, muon, tau (each with neutrino) → 6 types
• Quarks: Up, down, strange, charm, bottom, top → 6 types
• Spin: Up/down → 2 types
• Particle/antiparticle → 2 types
• Total: $(6+6)\times 2\times 2=48$

But considering color charge (quarks have 3 colors): $$\dim {\mathcal{H}}_{\text{fermion}}\sim 3\times 6\times 2\times 2=72$$

(Actually need more refined Fock space construction)

(2) Gauge Field Degrees of Freedom ${\mathcal{H}}_{\text{gauge}}$

Electromagnetic Field (U(1)):

• Photon: 2 polarization states
• Dimension: $\dim {\mathcal{H}}_{\text{gauge}}^{\text{EM}}=2$

Non-Abelian Gauge Fields (SU(N)):

• Gluons (SU(3) color gauge): 8 gluons × 2 polarizations = 16 states
• Weak gauge bosons (SU(2)): 3 bosons (W⁺, W⁻, Z) × 2 polarizations = 6 states

Combined (Standard Model SU(3)×SU(2)×U(1)): $$\dim {\mathcal{H}}_{\text{gauge}}\sim 16+6+2=24$$

(3) Auxiliary Qubits ${\mathcal{H}}_{\text{aux}}$

Why Needed: To ensure reversibility of QCA evolution.

Principle (Bennett garbage bits): Classical reversible computation needs “garbage registers” to store intermediate results, quantum QCA similar.

Dimension Estimate: If main degrees of freedom have $d_{\text{main}}$ states, auxiliary qubits usually need $d_{\text{aux}}\sim {\log }_2{d}_{\text{main}}$.

Standard Model QCA:

• Main degrees of freedom: $d_{\text{main}}\sim 72\times 24\sim 1700$
• Auxiliary qubits: $d_{\text{aux}}\sim {\log }_{2}1700\sim 11$ → $2^{11}=2048$

Total Cell Dimension: $$d_{\text{cell}}=d_{\text{fermion}}\times d_{\text{gauge}}\times d_{\text{aux}}\sim 72\times 24\times 2048\sim 3.5\times 10^6$$

(This is Hilbert space dimension of single lattice point!)

Encoding Cell Hilbert Space

Method 1 (Direct Encoding of Dimension):

• Store $d_{\text{cell}}$
• Need $\lceil {\log }_2{d}_{\text{cell}}\rceil$ bits
• Example: $d_{\text{cell}}=3.5\times 10^6$ → ${\log }_2(3.5\times 10^6)\approx 22$ bits

Method 2 (Decomposition Encoding):

• Separately store $d_{\text{fermion}},d_{\text{gauge}},d_{\text{aux}}$
• Total bits: ${\log }_2{d}_{\text{fermion}}+{\log }_2{d}_{\text{gauge}}+{\log }_2{d}_{\text{aux}}$
• Example: ${\log }_{2}72+{\log }_{2}24+{\log }_{2}2048=7+5+11=23$ bits

Method 3 (Specify Physical Model):

• Encode “Standard Model” (string)
• Dimension implicit in model
• Need: $\sim 50$ bits (encode model name+parameters)

Usually Choose: Method 3 (Physical model encoding)

Bit Count: $$I_{\mathcal{H}}\sim 50\text{ bits}$$

Tensor Product of Global Hilbert Space

Definition 3.4 (Global Hilbert Space):

$$\boxed{{\mathcal{H}}_{\Lambda }=\underset{x\in \Lambda }{⨂}{\mathcal{H}}_x}$$

Dimension: $$\dim {\mathcal{H}}_{\Lambda }=\prod _{x\in \Lambda }{d}_{\text{cell}}(x)=d_{\text{cell}}^{N_{\text{cell}}}$$

(Assuming cell dimensions same at all lattice points)

Numerical Example (Cosmological Scale):

• $d_{\text{cell}}=10^6$
• $N_{\text{cell}}=10^{90}$ (Observable universe in Planck units)
• $\dim {\mathcal{H}}_{\Lambda }=(10^6{)}^{10^{90}}=10^{6\times 10^{90}}$

This is a double exponential large number!

Maximum Entropy (Information Capacity): $$S_{\max }={\log }_2\dim {\mathcal{H}}_{\Lambda }=N_{\text{cell}}{\log }_2{d}_{\text{cell}}$$

Example:

• $N_{\text{cell}}=10^{90}$, $d_{\text{cell}}=10^6$
• $S_{\max }=10^{90}\times {\log }_2(10^6)=10^{90}\times 20=2\times 10^{91}$ bits

(Far exceeds $I_{\max }\sim 10^{123}$, meaning universe cannot “fill” entire Hilbert space!)

Part III: Boundary Conditions and Topology

Why Need Boundary Conditions?

Lattice set $\Lambda$ is finite, necessarily has “boundary”. How boundary is handled affects physical properties.

Classical Analogy:

• Open system: Energy can flow in/out (open boundary)
• Closed system: Energy conserved (periodic boundary)

Open Boundary Conditions

Definition 3.5 (Open Boundary):

Boundary lattice points only have partial neighbors (interior lattice points have normal number of neighbors).

One-Dimensional Example: $$\Lambda =\{0,1,2,\dots ,L-1\}$$

• Boundaries: $x=0,x=L-1$
• Interior: $x\in \{1,\dots ,L-2\}$

Neighbor Structure:

• $x=0$: Only right neighbor $x=1$
• $x=L-1$: Only left neighbor $x=L-2$
• $x\in \{1,\dots ,L-2\}$: Both left and right neighbors

Physical Meaning:

• Boundary is “real” (e.g., container wall)
• Quantum states can reflect or absorb at boundary
• Boundary effects significant (when $L$ not large enough)

Encoding:

• For each direction specify “open” → 1 bit/direction
• Total: $d$ bits

Periodic Boundary Conditions

Definition 3.6 (Periodic Boundary):

Boundary lattice points connect to opposite side through “wrapping”.

One-Dimensional Example: $$\Lambda ={\mathbb{Z}}_L=\{0,1,\dots ,L-1\}$$

Neighbor Structure (Nearest Neighbor):

• $x=0$: Left neighbor is $x=L-1$, right neighbor is $x=1$
• $x=L-1$: Left neighbor is $x=L-2$, right neighbor is $x=0$
• (Forms a “ring”)

Topology:

• One-dimensional periodic: Circle $S^1$
• Two-dimensional periodic: Torus ${\mathbb{T}}^2=S^1\times S^1$
• Three-dimensional periodic: Three-dimensional torus ${\mathbb{T}}^3$

Physical Meaning:

• Eliminates boundary effects
• Preserves translation symmetry
• Simulates “infinitely large” system (when $L$ large enough)

Encoding:

• For each direction specify “periodic” → 1 bit/direction
• Total: $d$ bits

Popular Analogy:

• Open boundary: Walking on flat map, stop at edge
• Periodic boundary: In game “Snake”, snake exits right side, re-enters from left

Twisted Boundary Conditions

Definition 3.7 (Twisted Boundary):

Apply a phase or symmetry transformation when wrapping.

One-Dimensional Example (Anti-Periodic): $$\psi (L)=-\psi (0)$$

(Wave function changes sign when wrapping)

Physical Meaning:

• Fermions: Usually use anti-periodic boundary (Pauli exclusion principle)
• Bosons: Use periodic boundary
• Topological phases: Need twisted boundary to detect topological invariants

Encoding:

• Specify twist type (none, anti-periodic, U(1) phase) → 2 bits/direction
• Total: $2d$ bits

Non-Trivial Topology

Example 1 (Three-Dimensional Sphere $S^3$):

• Closed, no boundary
• Need special lattice gluing

Example 2 (RP³, Manifolds):

• Complex topological invariants
• Need additional encoding of gluing maps

Encoding Overhead:

• Simple topology (${\mathbb{R}}^d$, ${\mathbb{T}}^d$, $S^d$): $\sim 10$ bits
• Complex topology (arbitrary manifolds): $\sim 100$ bits (Morse theory, CW complexes)

Cosmological Application:

• Observable universe topology unknown (may be ${\mathbb{R}}^3$, ${\mathbb{T}}^3$, hyperbolic space…)
• ${\Theta }_{\text{str}}$ needs to encode topology type

Bit Count of Boundary Conditions

$$I_{\text{boundary}}\sim 2d\text{ bits}$$

(Assuming standard or twisted periodic boundary)

Example ($d=3$): $$I_{\text{boundary}}=6\text{ bits}$$

Part IV: Symmetries and Conservation Laws

Why Are Symmetries Important?

Physical laws usually have symmetries:

• Time translation symmetry → Energy conservation
• Space translation symmetry → Momentum conservation
• Rotation symmetry → Angular momentum conservation
• Gauge symmetry → Charge conservation

In QCA framework, symmetries encoded in ${\Theta }_{\text{str}}$, affecting representation-theoretic structure of ${\mathcal{H}}_{\text{cell}}$.

Global Symmetry Group $G_{\text{global}}$

Definition 3.8 (Global Symmetry):

A unitary representation $\rho :G\to U({\mathcal{H}}_{\text{cell}})$ such that dynamics remains unchanged.

Example 1 (U(1) Symmetry):

• Particle number conservation
• Group: $G=U(1)=\{e^{i\theta }:\theta \in [0,2\pi )\}$
• Representation: $\rho (e^{i\theta })=e^{i\theta \hat{N}}$ ($\hat{N}$ is particle number operator)

Example 2 (SU(2) Spin Symmetry):

• Rotation invariance
• Group: $G=SU(2)$
• Representation: Spin-1/2, spin-1, etc.

Example 3 (Z₂ Symmetry):

• Parity symmetry ($x\to -x$)
• Group: $G={\mathbb{Z}}_2=\{+1,-1\}$
• Representation: $\rho (-1)=P$ (parity operator)

Local Gauge Symmetry $G_{\text{local}}$

Definition 3.9 (Gauge Symmetry):

Symmetry transformations acting independently at each lattice point, physical states equivalent under gauge transformations.

Standard Model: $$G_{\text{local}}=SU(3{)}_{\text{color}}\times SU(2{)}_{\text{weak}}\times U(1{)}_Y$$

• SU(3): Color gauge symmetry (strong interaction)
• SU(2): Weak isospin symmetry
• U(1): Hypercharge symmetry

Physical Hilbert Space: Need states satisfying Gauss law (gauge constraints).

Example (Lattice Gauge Theory):

• Place gauge field variables $U_e\in SU(N)$ on each edge
• Physical states satisfy: ${\prod }_{e\text{ out of }x}{U}_e=1$ (at each lattice point)

Encoding Symmetries

Encoded Content:

1. Symmetry Group Type:

   • “U(1)”, “SU(2)”, “SU(3)”, …
   • Use string or enumeration type → $\sim 20$ bits

2. Representation Choice:

   • Fundamental representation, adjoint representation, spin-$j$ representation…
   • Each representation $\sim 10$ bits

3. How Acts on ${\mathcal{H}}_{\text{cell}}$:

   • Specify matrix representation of generators on basis vectors
   • For standard groups (U(1), SU(2), SU(3)), can preset
   • Additional encoding $\sim 10$ bits

Total Bits: $$I_{\text{symmetry}}\sim 20+10+10=40\text{ bits}$$

(For single symmetry group)

Standard Model (SU(3)×SU(2)×U(1)): $$I_{\text{symmetry}}\sim 3\times 40=120\text{ bits}$$

Symmetries and Information Compression

Key Insight: Symmetries can greatly compress parameter information!

Example (Translation Invariance):

Without Symmetry:

• Hamiltonian parameters at each lattice point independent → $N_{\text{cell}}$ sets of parameters
• Information: $\sim N_{\text{cell}}\times I_{\text{local}}$

With Translation Symmetry:

• All lattice point Hamiltonians same → Only need 1 set of parameters
• Information: $\sim I_{\text{local}}$

Compression Ratio: $N_{\text{cell}}$ (astronomical number!)

Physical Necessity: If universe had no symmetry, parameter information needed would exceed $I_{\max }$ → Contradiction

Therefore: Symmetry is not coincidence, but necessary consequence of finite information axiom!

Part V: Defects and Non-Uniform Structure

Topological Defects

Definition 3.10 (Topological Defect):

Positions in space where field configuration or geometry has singularity, cannot be eliminated by continuous deformation.

Example 1 (Cosmic String):

• One-dimensional defect (“string” in three-dimensional space)
• Field configuration gains non-trivial phase when circling string once
• Produces conical geometry (deficit angle)

Example 2 (Magnetic Monopole):

• Zero-dimensional defect (“point”)
• Gauge field at infinity similar to magnetic monopole field
• Dirac quantization condition

Example 3 (Domain Wall):

• Two-dimensional defect (“wall” in three-dimensional space)
• Vacuum states on two sides different (spontaneous symmetry breaking)

Encoding Defects in QCA

Method 1 (Position List):

• Defect positions: $(x_1,y_1,z_1),(x_2,y_2,z_2),\dots$
• Each coordinate $\sim 3\times {\log }_{2}L$ bits
• $k$ defects: $\sim k\times 3{\log }_{2}L$ bits

Method 2 (Field Configuration Encoding):

• Near defects, field configuration special
• Encode defect type (string, monopole, domain wall) + orientation
• Each defect $\sim 50$ bits

Cosmology (Post-Inflation Remnants):

• Inflation theory predicts: Universe may have $\sim 0$ large-scale defects (diluted by inflation)
• Or $\sim 10$ defects (formed during phase transition)

Encoding Overhead: $$I_{\text{defect}}\sim k_{\text{defect}}\times 50\text{ bits}$$

Example ($k_{\text{defect}}=0$): $$I_{\text{defect}}=0\text{ bits}$$

(Uniform universe, no defects)

Non-Uniform Lattice Lengths (Refinement)

Motivation: Some regions need higher resolution.

Example (Astronomy):

• Near galaxy clusters: High resolution ($a\sim 10^{-10}{\ell }_p$)
• Interstellar space: Low resolution ($a\sim 10^{-5}{\ell }_p$)

Implementation (Adaptive Grid):

• “Subdivide” some coarse lattice points into $2^d$ sub-lattice points
• Recursive subdivision

Encoding:

• Subdivision tree structure (like quadtree/octree)
• Each subdivision decision $\sim 1$ bit
• If subdivide $k_{\text{refine}}$ times: $\sim k_{\text{refine}}$ bits

Cosmological Application:

• Observations show universe large-scale uniform (CMB fluctuations $\sim 10^{-5}$)
• Non-uniform structure mainly at small scales (galaxies, stars)
• If use coarse-graining, small scales emerge
• ${\Theta }_{\text{str}}$ can only encode large-scale uniform lattice

Typical Value: $$I_{\text{refine}}\sim 0\text{ bits}$$

(Uniform lattice sufficient)

Part VI: Total Bit Count of Structural Parameters

Combining above parts:

$$\boxed{|{\Theta }_{\text{str}}|=I_{\Lambda }+I_{\mathcal{H}}+I_{\text{boundary}}+I_{\text{symmetry}}+I_{\text{defect}}+I_{\text{refine}}}$$

Numerical Table (Standard Universe QCA):

Key Observation: $$|{\Theta }_{\text{str}}|\sim 400\text{ bits}\ll I_{\max }\sim 10^{123}\text{ bits}$$

Structural parameter information extremely small!

Part VII: Construction of Quasi-Local $C^{\ast }$ Algebra

From Lattice Points to Algebra

With lattice set $\Lambda$ and cell Hilbert space ${\mathcal{H}}_x$, can construct quasi-local operator algebra.

Definition 3.11 (Local Algebra):

For finite subset $F\subset \Lambda$, define:

$${\mathcal{H}}_F=\underset{x\in F}{⨂}{\mathcal{H}}_x$$

$${\mathcal{A}}_F=\mathcal{B}({\mathcal{H}}_F)$$

(All bounded operators on ${\mathcal{H}}_F$)

Embedding:

If $F\subset F^{\prime }$, then ${\mathcal{A}}_F\subset {\mathcal{A}}_{F^{\prime }}$ through:

$${\iota }_{F\subset F^{\prime }}(A_F)=A_F\otimes 1_{F^{\prime }\setminus F}$$

(Act as identity outside $F$)

Definition 3.12 (Quasi-Local $C^{\ast }$ Algebra):

$$\boxed{\mathcal{A}(\Lambda )={\overline{\bigcup _{F⋐\Lambda }{\mathcal{A}}_F}}^{\Vert \cdot \Vert }}$$

(Closure of all local operators, in operator norm)

Physical Meaning:

• $\mathcal{A}(\Lambda )$: All “local observables”
• Observables act on finite regions, but can be arbitrarily large

State Space

Definition 3.13 (State):

State is positive, normalized linear functional on $\mathcal{A}(\Lambda )$:

$$\omega :\mathcal{A}(\Lambda )\to \mathbb{C}$$

Satisfying:

• Positivity: $\omega (A^{†}A)\ge 0$
• Normalization: $\omega (1)=1$

Pure and Mixed States:

• Pure state: $\omega (A)=⟨\psi |A|\psi ⟩$ (some vector state)
• Mixed state: $\omega (A)=\text{tr}(\rho A)$ (density matrix)

State Space Dimension: $$\dim (\text{pure state manifold})=2\dim {\mathcal{H}}_{\Lambda }-2=2d_{\text{cell}}^{N_{\text{cell}}}-2$$

(Real dimension of complex projective space ${\mathbb{CP}}^{d-1}$)

This is double exponential large!

Relationship Between $C^{\ast }$ Algebra and Finite Information

Key Theorem:

On finite-dimensional ${\mathcal{H}}_{\Lambda }$, $\mathcal{A}(\Lambda )=\mathcal{B}({\mathcal{H}}_{\Lambda })$ is also finite-dimensional:

$${\dim }_{\mathbb{C}}\mathcal{A}(\Lambda )=(\dim {\mathcal{H}}_{\Lambda }{)}^2=d_{\text{cell}}^{2N_{\text{cell}}}$$

Information Content: $${\log }_2\dim \mathcal{A}(\Lambda )=2N_{\text{cell}}{\log }_2{d}_{\text{cell}}=2S_{\max }$$

But number of physically observable operators far less than $\dim \mathcal{A}$, because:

• Symmetry constraints (gauge invariant, translation invariant)
• Locality (experiments can only measure local operators)

Effective Information: $$I_{\text{eff}}^{\text{obs}}\ll 2S_{\max }$$

(Usually $\sim S_{\max }$ or less)

Summary of Core Points of This Article

Five Components of Structural Parameters

$${\Theta }_{\text{str}}=(\Lambda ,{\mathcal{H}}_{\text{cell}},\text{boundary},G_{\text{symm}},\text{defects})$$

Global Hilbert Space

$${\mathcal{H}}_{\Lambda }=\underset{x\in \Lambda }{⨂}{\mathcal{H}}_x,\dim {\mathcal{H}}_{\Lambda }=d_{\text{cell}}^{N_{\text{cell}}}$$

Maximum Entropy: $$S_{\max }=N_{\text{cell}}{\log }_2{d}_{\text{cell}}$$

Numerical Example (Cosmological):

• $N_{\text{cell}}\sim 10^{90}$, $d_{\text{cell}}\sim 10^6$
• $S_{\max }\sim 2\times 10^{91}$ bits

Quasi-Local $C^{\ast }$ Algebra

$$\mathcal{A}(\Lambda )={\overline{\bigcup _{F⋐\Lambda }\mathcal{B}({\mathcal{H}}_F)}}^{\Vert \cdot \Vert }$$

Physical Meaning: Set of all local observables.

Core Insights

1. Structural parameters tiny: $|{\Theta }_{\text{str}}|\sim 400\ll I_{\max }\sim 10^{123}$
2. State space huge: $S_{\max }\sim 10^{91}$ dominates information capacity
3. Symmetry necessary: No symmetry → Parameter explosion → Exceeds $I_{\max }$
4. Finite information forces discreteness: Continuous spacetime needs infinite information → Must discretize
5. Lattice spacing $a$ and physical scale: $a\sim {\ell }_p$ (Planck length) is natural unit

Relationship with Continuous Field Theory

Continuous Limit (Article 07 will detail): $$a,\Delta t\to 0\Rightarrow \text{QCA}\to \text{Field Theory}$$

Next Article Preview: 04. Detailed Explanation of Dynamical Parameters: Source Code of Physical Laws

• Construction of QCA automorphism ${\alpha }_{\Theta }$
• Finite depth local unitary circuits
• Gate set $\mathcal{G}$ and universality
• Discrete angle parameters $\theta \to 2\pi n/2^m$
• Lieb-Robinson bound and light cone
• From discrete gates to continuous Hamiltonian